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A triangular array related to A077028 and distributing the values of A007582.
1

%I #19 Jan 24 2019 05:47:53

%S 1,1,2,1,5,4,1,10,17,8,1,19,51,49,16,1,36,134,196,129,32,1,69,330,650,

%T 645,321,64,1,134,783,1940,2575,1926,769,128,1,263,1813,5411,8995,

%U 8981,5383,1793,256,1,520,4124,14392,28742,35896,28700,14344,4097,512,1,1033,9252,36948,86142,129150,129108,86052,36873,9217,1024

%N A triangular array related to A077028 and distributing the values of A007582.

%C Let T(r,c) be the array A077028. Fill 2^k numbers in Gaussian templates conforming to the row lengths determined by T(r,c). A110552 results from summing the numbers on each row.

%H G. C. Greubel, <a href="/A110552/b110552.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%H P. Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Barry2/barry231.html">A Note on a Family of Generalized Pascal Matrices Defined by Riordan Arrays</a>, Journal of Integer Sequences, 16 (2013), #13.5.4.

%F Table entries appear to be given by T(n,k) = binomial(n-2,k-1) + 2^(n-1)*binomial(n-2,k-2), n,k >= 1, leading to the e.g.f. (exp((1+x)*u) - 1)*(x*exp((1+x)*u) + x + 2)/(2*(1+x)^2) = u + (1+2*x)*u^2/2! + (1+5*x+4*x^2)*u^3/3! + .... Cf. A111049. - _Peter Bala_, Jul 27 2012

%e The filled templates begin

%e 1

%e .1

%e .2

%e ..1

%e ..2.3

%e ..4

%e ....1

%e ....2.3.5

%e ....4.6.7

%e ....8

%e therefore the sequence begins

%e 1

%e 1 2

%e 1 5 4

%e 1 10 17 8

%e ...

%t T[n_, k_] := Binomial[n - 2, k - 1] + 2^(n - 1)*Binomial[n - 2, k - 2]; Table[T[n, k], {n, 1, 20}, {k, 1, n}] // Flatten (* _G. C. Greubel_, Aug 31 2017 *)

%o (PARI) for(n=1,20, for(k=1,n, print1(binomial(n - 2, k - 1) + 2^(n - 1)*binomial(n - 2, k - 2), ", "))) \\ _G. C. Greubel_, Aug 31 2017

%Y Cf. A077028, A007051, A007582. A111049.

%K nonn,tabl

%O 1,3

%A _Alford Arnold_, Jul 26 2005